ABSTRACT
The Hamiltonian for the many-electron system in finite volume [0, L]d is given by H = H0 + Hint where
H0 = 1Ld ∑
eka + kσakσ (6.1)
ek = k 2
2m − µ, and
Hint = 1L3d ∑
∑
V (k− p)a+kσa+q−k,τaq−p,τapσ (6.2)
As usual, the interacting part may be represented by the following diagram:
Since there is conservation of momentum, there are three independent momenta. One may consider three natural limiting cases with two independent momenta:
forward exchange BCS
In chapter 8 where we prove rigorous bounds on Feynman diagrams we show that the most singular contributions to the perturbation series come from the
the third diagram in the above figure. To retain only this term, the q = 0 term of Hint given by (6.2), is the basic approximation of BCS theory. Furthermore V (k−p) is substituted by its value on the Fermi surface, V (k − p) ≈ V (kˆ − pˆ) where kˆ = kF k‖k‖ , kF =
√ 2mµ, and the momenta k,p are restricted to values close to the
Fermi surface, |ek|, |ep| ≤ ωD, where the cutoff ωD is referred to as the Debye frequency. Thus the BCS approximation reads
Hint ≈ HBCS (6.3) where
HBCS = 1L3d ∑
∑
V (kˆ− pˆ) a+kσa+−k,τa−p,τapσ (6.4)
We first make a comment on the volume factors. In (6.4) we retained only the q = 0 term of (6.2), but we did not cancel a volume factor. It is not obvious that (6.4) is still proportional to the volume. However, in the case of the forward approximation which is defined by putting k = p in (6.2) without canceling a volume factor,
Hint ≈ Hforw := 1L3d ∑
∑
V (0)a+kσa + p,τap,τakσ (6.5)
there is an easy argument which shows that (6.5) is still proportional to the volume (for constant density). Namely, on a fixed N -particle space FN the interacting part Hint is a multiplication operator given by
N∑
V (xi − xj) (6.6)
Let δy(x) = δ(x − y). Then ϕ(x1, · · · ,xN ) := δy1 ∧ · · · ∧ δyN (x1, · · · ,xN ) is an eigenfunction of Hint with eigenvalue
E = 12
N∑
V (yi − yj) = 12Ld N∑
∑
e−i(yi−yj)qV (q) (6.7)
which is, for V (x) ∈ L1, proportional to N or to the volume for constant density. One finds that ϕ is also an eigenvector of the forward term, Hforwϕ = Eforwϕ where Eforw is obtained from (6.7) by putting q = 0 without canceling a volume factor,
V (q = 0) (6.8)
Although the model defined by the Hamiltonian HBCS in (6.4) is still quartic in the annihilation and creation operators, it can be solved explicitly, without making a quadratic mean field approximation as it is usually done. The important point is that the approximation ‘putting q = 0 without canceling a volume factor’ has the effect that in the bosonic functional integral representation the volume factors enter the formulae in such a way that in the thermodynamic limit the integration variables are forced to take values at the global minimum of the effective potential. That is, fluctuations around the minimum configuration are suppressed for Ld →∞.
